It is known from the spectral theory [1]}, [2]}, [3]} that the linear operator \(-\frac{\partial _z^2}{2}+ (U_{\rho _{(\mathbf {f}^*,\chi ^*)}}+V_{\textup {ext}})(y,\cdot )\)  has only simple eigenvalues so that \(\lambda _j^*(y)\)  is strictly increasing in \(j\ge 1\) . Moreover, \(\lambda _j^*(y)\)  has a lower bound \(\ge \frac{(\pi j)^2}{3}\)  (following the proof of (REF )). Therefore, for every \((y,v)\in \omega \times \mathbb {R}^2\) , \(f_j(y,v)\)  is strictly decreasing in \(j\)  and \(f_j^*(y,v)\equiv 0\)  for \(j\ge \frac{\sqrt{3\mu }}{\pi }\) .
